On the Satisfiability Threshold and Clustering of Solutions of Random 3-SAT Formulas

TL;DR

This paper investigates the structure of solutions in random 3-SAT formulas, showing that above a certain density, core assignments almost surely do not exist, which impacts understanding of solution clustering and the satisfiability threshold.

Contribution

It demonstrates that for densities above 4.453, random 3-SAT formulas almost surely lack core assignments, providing new insights into the solution space structure and thresholds.

Findings

No non-trivial core assignments above density 4.453

Implication that the satisfiability threshold is below 4.453 or cores do not exist in 3-SAT

Introduces a simple first moment method application for analysis

Abstract

We study the structure of satisfying assignments of a random 3-SAT formula. In particular, we show that a random formula of density 4.453 or higher almost surely has no non-trivial "core" assignments. Core assignments are certain partial assignments that can be extended to satisfying assignments, and have been studied recently in connection with the Survey Propagation heuristic for random SAT. Their existence implies the presence of clusters of solutions, and they have been shown to exist with high probability below the satisfiability threshold for k-SAT with k>8, by Achlioptas and Ricci-Tersenghi, STOC 2006. Our result implies that either this does not hold for 3-SAT or the threshold density for satisfiability in 3-SAT lies below 4.453. The main technical tool that we use is a novel simple application of the first moment method.